Math Lesson 16.2.4 - Bijective Function

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Welcome to our Math lesson on Bijective Function, this is the fourth lesson of our suite of math lessons covering the topic of Injective, Surjective and Bijective Functions. Graphs of Functions, you can find links to the other lessons within this tutorial and access additional Math learning resources below this lesson.

Bijective Function

By definition, a bijective function is a type of function that is injective and surjective at the same time. In other words, a surjective function must be one-to-one and have all output values connected to a single input.

For example, all linear functions defined in R are bijective because every y-value has a unique x-value in correspondence.

We can define a bijective function in a more formal language as follows:

A function f(x) (from set X to Y) is bijective if, for every y in Y, there is exactly one x in X such that f(x) = y.

Some functions may be bijective in one domain set and bijective in another. For example, f(x) = x^x is not bijective in R because it is not injective. However, if the domain is restricted to the set of natural numbers N, then this function becomes bijective because there are no more negative inputs, which may give the same output as their positive correspondent.

Example 4

1. What is the domain of the function f(x) = √54 - 6x² ?
2. What is the range of this function?
3. What kind of function (injective, surjective or bijective) is this function in its domain?
4. How can you make it bijective?

Solution 4

1. The function is determined for f(x) ≥ 0. This means there must be 54 - 6x² ≥ 0. Thus,
   54-6x² ≥ 0
   54 ≥ 6x²
   x² ≤ 54/6
   x² ≤ 9
   This means x ≥ -3 and x ≤ 3. In other words, the domain is D = [-3, 3].
2. The range R extends between the minimum and maximum value of this function. The minimum value is obtained for x = -3 or 3 and it is y_min = f(-3) = f(3) = √[54 - 6 · (-3)²] = √[54 - 6 · 32] = 0, while the maximum value is obtained for x = 0, i.e. y_max = f(0) = √(54 - 6 · 0²) = √54.Therefore, the range of this function is R = [0, 54].
3. This is not an injective function as for two different inputs (for example, x = -2 and x = 2) we obtain the same output f(x) = f(-x) = √30.
   However, this is a surjective function, as for every y-value from the range, there is at least one x-value from the domain. (In general, there are two x-values for a single y-value except for x = 0, which is unique.)
   For this reason, the given function can't be bijective, as it does not meet both the necessary requirements.
4. We can make this function bijective by restricting the domain to only one of its halves, i.e. from -3 to 0 or from 0 to 3.

You have reached the end of Math lesson 16.2.4 Bijective Function. There are 7 lessons in this physics tutorial covering Injective, Surjective and Bijective Functions. Graphs of Functions, you can access all the lessons from this tutorial below.

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